Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:math-ph/9910036

The analytic quantum information manifold

R. F. Streater

Published Oct 22, 1999• 10 citations
arXivHTMLPDF

Abstract

Let H be a self-adjoint operator such that exp(-aH) is of trace class for some a<1. Let V be a symmetric operator, Kato bounded relative to H. We show that log Tr[exp(-H+xV)] is a real analytic function of x in a hood of x=0. We show that the Gibbs states of H+xV form a real analytic Banach manifold. This work has been extended in math-ph/9910031.

DOI
10.48550/arxiv.math-ph/9910036

Related papers

The information manifold for relatively bounded forms

R. F. Streater

We construct a Banach manifold of states, which are Gibbs states for potentials that are form-bounded in the sense of Kato relative to the free Hamiltonian. We construct the (+1)-affine structure and the (+1)-affine connection in the sense of Amari.

arXiv:math-ph/9910035•Oct 22, 1999•2 citations

Quantum Orlicz Spaces in Information Geometry

R. F. Streater

Let H 0 be a selfadjoint operator such that Tr e −βH 0 is of trace class for some β < 1, and let χ ɛ denote the set of ɛ-bounded forms, i.e., ∥(H 0 +C) −1/2−ɛ X(H 0 +C) −1/2+ɛ ∥ < C for some C > 0. Let χ := Span ∪ ɛ∈(0,1/2] χ ɛ . Let [Formula: see text] denote the underlying set of the quantum information manifold of states of the form ρ x = e −H 0 −X−ψx , X ∈ χ. We show that if Tr e −H 0 = 1. 1. the map Φ, [Formula: see text] is a quantum Young function defined on χ 2. The Orlicz space defined by Φ is the tangent space of [Formula: see text] at ρ 0 ; its affine structure is defined by the (+1)-connection of Amari 3. The subset of a ‘hood of ρ 0 , consisting of p-nearby states (those [Formula: see text] obeying C −1 ρ 1+p ≤ σ ≤ Cρ 1 − p for some C > 1) admits a flat affine connection known as the (−1) connection, and the span of this set is part of the cotangent space of [Formula: see text] 4. These dual structures extend to the completions in the Luxemburg norms.

Open Systems & Information Dynamics•Dec 1, 2004•52 citations

On ergodic limits of normal states in quantum statistical mechanics

Eduard Prugovečki, A. Tip

The asymptotic behavior for t→±∞ of S(t)=exp(−i H t)Sexp(+i H t) and its time average S̄ (t)=t−1∫0tdu S(u) is discussed. Here is S an element of the Banach space B1, constituted by the trace class of operators on the (separable or nonseparable) Hilbert space H, and H is the Hamiltonian, i.e., a bounded or unbounded self-adjoint operator on H. Necessary and sufficient conditions are given for the existence of the limits S̄(± ∞) and S(± ∞) with respect to the weak topology on B1, for the latter under the assumption that the continuous spectrum of H is absolutely continuous. In addition it is shown that if, for a normal state (density operator) ρ, ρ̄(t) has a weak limit, then the limit is again a normal state. This provides further insight in the nature of Emch's ``first ergodic paradox'' [G. G. Emch, J. Math. Phys. 7, 1413 (1966)].

Journal of Mathematical Physics•Mar 1, 1974•7 citations

Analysis of generalized Gibbs states

Stephan Weis, Ilya M. Spitkovsky

An exponential family is a manifold of generalized Gibbs state of the form exp(H)/Tr(exp(H)), where H belongs to a vector space of (possibly non-commutative) hermitian matrices. Generalized Gibbs states are important in small-scale thermodynamics, they represent equilibrium states regarding several conserved quantities that admit novel operations without heat dissipation [1]. Quantum information theory and condensed matter physics consider a space of local Hamiltonians acting on spins. The entropy distance from this exponential family is a measure of many-body complexity [2,3,4]. This talk is concerned with the geometry and topology of an exponential family and its entropy distance [5]. The maximum-entropy inference map parametrizes the exponential family. This map is continuous in the interior of its domain, the joint numerical range [6]. We describe the points of discontinuity in terms of open mapping theorems and eigenvalue crossings. Because of the discontinuity, the inference map and the entropy distance cannot be approximated through interior points. Instead, it is necessary to study faces (flat portions on the boundary) of the joint numerical range. With local Hamiltonians, this requires studying the faces of the set of quantum marginals. References. [1] M. Lostaglio, D. Jennings, and T. Rudolph,New Journal of Physics 19, 043008 (2017). [2] N. Ay, E. Olbrich, N. Bertschinger, and J. Jost,Chaos 21, 037103 (2011). [3] S. Niekamp, T. Galla, M. Kleinmann, and O. Gühne, J. Phys. A-Math. Gen. 46, 125301 (2013). [4] B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen, Quantum Information Meets Quantum Matter,New York: Springer, 2019. [5] S. Weis, Journal of Convex Analysis 21, 339-399 (2014). [6] L. Rodman, I. M. Spitkovsky, A. Szkoła, and S. Weis,Journal of Mathematical Physics 57, 015204 (2016).

Proceedings of Entropy 2021: The Scientific Tool of the 21st Century•May 5, 2021•0 citations

Duality in quantum information manifolds

R. F. Streater

We consider the quantum information manifold whose underlying set M consists of density operators rho with the extra property that some fractional power of rho is of trace class. The topology is defined by defining a neighbourhood of a point rho to be all density operators that dominate rho and are dominated by rho. We show that this is the same set as that of all states whose relative Hamiltonian X in the sense of Araki is bounded, and such that X(t) is holomorphic in the circle |t| less than 1/2. Here, X(t) is the time evolution of X determined by the modular automorphism defined by rho. We show that M is a Banach manifold in Araki's norm, and that both the canonical and the mixture affine connections can be defined. These are dual relative to the Kubo-Mori metric, and so generalise Amari's dual theory to quantum theory in infinite dimensions.

arXiv:math-ph/0308037•Aug 28, 2003•1 citation

Convexity inequalities for estimating free energy and relative entropy

Mary Beth Ruskai, Frank H. Stillinger

For quantum systems, it is shown that the relative entropy S(P,Q)=-Tr P log P+Tr P log Q of two positive semi-definite operators P and Q satisfies gamma -1 Tr(P-P 1+ gamma Q - gamma )<or=S(P,Q)<or= gamma -1 Tr(P 1- gamma Q gamma -P) for 0( gamma <or=1, and that these bounds become exact in the limit gamma to Phi . Analogous inequalities hold, for states in classical statistical mechanics or information theory, with trace replaced by integration or summation. Furthermore, the average of these bounds is, in general a better approximation to S(P,Q) than either bound alone, and the average is amenable to further improvement via repeated Richardson extrapolation. IF P and Q are Gibbs equilibrium states, then these inequalities can also be used to obtain bounds on the free energy of a perturbed system in terms of the free energy of the unperturbed system and expectations of the perturbation.

Journal of Physics A Mathematical and General•Jun 21, 1990•38 citations
Let H be a self-adjoint operator such that exp(-aH) is of trace class for some a<1. Let V be a symmetric operator, Kato bounded relative to H. We show that log Tr[exp(-H+xV)] is a real analytic function of x in a hood of x=0. We show that the Gibbs states of H+xV form a real analytic Banach manifold. This work has been extended in math-ph/9910031.
Let H 0 be a selfadjoint operator such that Tr e −βH 0 is of trace class for some β < 1, and let χ ɛ denote the set of ɛ-bounded forms, i.e., ∥(H 0 +C) −1/2−ɛ X(H 0 +C) −1/2+ɛ ∥ < C for some C > 0. Let χ := Span ∪ ɛ∈(0,1/2] χ ɛ . Let [Formula: see text] denote the underlying set of the quantum information manifold of states of the form ρ x = e −H 0 −X−ψx , X ∈ χ. We show that if Tr e −H 0 = 1. 1. the map Φ, [Formula: see text] is a quantum Young function defined on χ 2. The Orlicz space defined by Φ is the tangent space of [Formula: see text] at ρ 0 ; its affine structure is defined by the (+1)-connection of Amari 3. The subset of a ‘hood of ρ 0 , consisting of p-nearby states (those [Formula: see text] obeying C −1 ρ 1+p ≤ σ ≤ Cρ 1 − p for some C > 1) admits a flat affine connection known as the (−1) connection, and the span of this set is part of the cotangent space of [Formula: see text] 4. These dual structures extend to the completions in the Luxemburg norms.
The asymptotic behavior for t→±∞ of S(t)=exp(−i H t)Sexp(+i H t) and its time average S̄ (t)=t−1∫0tdu S(u) is discussed. Here is S an element of the Banach space B1, constituted by the trace class of operators on the (separable or nonseparable) Hilbert space H, and H is the Hamiltonian, i.e., a bounded or unbounded self-adjoint operator on H. Necessary and sufficient conditions are given for the existence of the limits S̄(± ∞) and S(± ∞) with respect to the weak topology on B1, for the latter under the assumption that the continuous spectrum of H is absolutely continuous. In addition it is shown that if, for a normal state (density operator) ρ, ρ̄(t) has a weak limit, then the limit is again a normal state. This provides further insight in the nature of Emch's ``first ergodic paradox'' [G. G. Emch, J. Math. Phys. 7, 1413 (1966)].
An exponential family is a manifold of generalized Gibbs state of the form exp(H)/Tr(exp(H)), where H belongs to a vector space of (possibly non-commutative) hermitian matrices. Generalized Gibbs states are important in small-scale thermodynamics, they represent equilibrium states regarding several conserved quantities that admit novel operations without heat dissipation [1]. Quantum information theory and condensed matter physics consider a space of local Hamiltonians acting on spins. The entropy distance from this exponential family is a measure of many-body complexity [2,3,4]. This talk is concerned with the geometry and topology of an exponential family and its entropy distance [5]. The maximum-entropy inference map parametrizes the exponential family. This map is continuous in the interior of its domain, the joint numerical range [6]. We describe the points of discontinuity in terms of open mapping theorems and eigenvalue crossings. Because of the discontinuity, the inference map and the entropy distance cannot be approximated through interior points. Instead, it is necessary to study faces (flat portions on the boundary) of the joint numerical range. With local Hamiltonians, this requires studying the faces of the set of quantum marginals. References. [1] M. Lostaglio, D. Jennings, and T. Rudolph,New Journal of Physics 19, 043008 (2017). [2] N. Ay, E. Olbrich, N. Bertschinger, and J. Jost,Chaos 21, 037103 (2011). [3] S. Niekamp, T. Galla, M. Kleinmann, and O. Gühne, J. Phys. A-Math. Gen. 46, 125301 (2013). [4] B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen, Quantum Information Meets Quantum Matter,New York: Springer, 2019. [5] S. Weis, Journal of Convex Analysis 21, 339-399 (2014). [6] L. Rodman, I. M. Spitkovsky, A. Szkoła, and S. Weis,Journal of Mathematical Physics 57, 015204 (2016).
We consider the quantum information manifold whose underlying set M consists of density operators rho with the extra property that some fractional power of rho is of trace class. The topology is defined by defining a neighbourhood of a point rho to be all density operators that dominate rho and are dominated by rho. We show that this is the same set as that of all states whose relative Hamiltonian X in the sense of Araki is bounded, and such that X(t) is holomorphic in the circle |t| less than 1/2. Here, X(t) is the time evolution of X determined by the modular automorphism defined by rho. We show that M is a Banach manifold in Araki's norm, and that both the canonical and the mixture affine connections can be defined. These are dual relative to the Kubo-Mori metric, and so generalise Amari's dual theory to quantum theory in infinite dimensions.
For quantum systems, it is shown that the relative entropy S(P,Q)=-Tr P log P+Tr P log Q of two positive semi-definite operators P and Q satisfies gamma -1 Tr(P-P 1+ gamma Q - gamma )<or=S(P,Q)<or= gamma -1 Tr(P 1- gamma Q gamma -P) for 0( gamma <or=1, and that these bounds become exact in the limit gamma to Phi . Analogous inequalities hold, for states in classical statistical mechanics or information theory, with trace replaced by integration or summation. Furthermore, the average of these bounds is, in general a better approximation to S(P,Q) than either bound alone, and the average is amenable to further improvement via repeated Richardson extrapolation. IF P and Q are Gibbs equilibrium states, then these inequalities can also be used to obtain bounds on the free energy of a perturbed system in terms of the free energy of the unperturbed system and expectations of the perturbation.